All Questions
Tagged with c-star-algebras gr.group-theory
32 questions
64
votes
4
answers
8k
views
What is the current status of the Kaplansky zero-divisor conjecture for group rings?
Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group.
A couple of good ...
- 1,065
20
votes
2
answers
870
views
C$^*$-algebras isomorphic after tensoring with $M_n(\mathbb C)$
In 1977, Joan Plastiras gave a striking example of two non $*$-isomorphic C$^*$-algebras $\mathcal A$ and $\mathcal B$ such that $$\mathcal A \otimes M_2(\mathbb C) \simeq \mathcal B\otimes M_2(\...
- 3,984
19
votes
0
answers
474
views
On C*-rigidity problem for torsion-free groups
I'd like to address the $\mathrm{C}^\ast$-rigidity problem for
torsion-free groups (see
this paper),
which asks for non-isomorphic torsion-free groups with isomorphic
(reduced) group $\mathrm{C}^\ast$-...
- 10.1k
18
votes
2
answers
925
views
Which groups are the unitary group of a $C^*$-algebra
Which groups are the unitary group of a $C^*$-algebra?
Does anyone know anything in this direction?
- 221
12
votes
1
answer
553
views
Topological amenability vs amenability of an action
Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms.
Consider the following two definitions:
[$C^*$-algebras and finite dimensional ...
- 121
11
votes
2
answers
636
views
Quasinilpotent elements of group C-star algebras
If $G$ is a discrete torsion-free group, can its (reduced or full) group C-star algebra contain non-zero quasinilpotent elements? I've seen various examples in the group von Neumann algebra setting (...
- 25.8k
11
votes
0
answers
378
views
Why are projectionless $C^*$-algebras important (Kadison's conjecture)
It was considered an important result for a long time to show that the reduced $C^*$-algebra of the free group $C^*_r(F_2)$ has no nontrivial projections. I believe this is also known as Kadison's ...
- 211
10
votes
2
answers
1k
views
Kazhdan's property (T) vs. residual finiteness
I have asked this question already on mathstackexchange but got no answer (see https://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...
- 721
9
votes
3
answers
453
views
Comparison between the operator norm and the $L^1$ norm on group algebras
Consider a discrete group $G$ and its group algebra over $\mathbb{C}$, $\mathbb{C}[G]$. There are four norms on it I wish to consider for this question:
The 2-norm given by $||\sum_{g \in G} c_gg||_2^...
- 2,830
9
votes
1
answer
372
views
Simplicity of group $C^\ast$-algebra implies fullness of group-von Neumann algebra?
Let $\Gamma$ be a discrete group whose reduced group $C^\ast$-algebra is simple. Can we conclude that the corresponding group-von Neumann algebra $\mathcal{L}(G)$ is a full $\text{II}_1$-factor, ...
- 741
9
votes
2
answers
2k
views
What does the representation theory of the reduced C*-algebra correspond to?
Let $G$ be a locally compact group. The group C*-algebra $C^* (G)$ is designed to come with a natural bijection between its (nondegenerate) representations and the (strongly continuous, unitary) ...
- 3,174
9
votes
3
answers
2k
views
Conjugacy classes and reduced group $C^*$-algebra of an amenable group
The reduced $C^*$-algebra of a non-abelian free group $G$ has a unique trace. Hence, there is no chance to separate conjugacy classes of group elements using traces on $C^\star_{red} G$. On the other ...
- 25.5k
7
votes
2
answers
871
views
Amenable action intuition
Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra. Consider an action $\alpha: \Gamma \to \operatorname{Aut}(A)$. There is a notion of amenability for such an action (see e.g. Brown and ...
- 175
7
votes
1
answer
476
views
How can one define a kind of "determinant" on a reduced group $C^*$ algebra?
Let $A$ be a unital $C^*$-algebra which is equipped with a faithful trace $T$. In particular we may consider $A=C^*_{\text{red}} (G)$ for some discrete group $G$. We consider the following ...
- 356
6
votes
2
answers
711
views
maximal tensor product of simple $C^*$algebras is non-simple
Let $A$ and $B$ simple $C^*$-algebras. One can prove that the minimal tensor product $A\otimes _{min}B$ is simple. This is wrong for the maximal tensor product $A\otimes_{max}B$ .
1.Do you know an ...
- 1,188
5
votes
1
answer
622
views
Can the full and reduced group $C^*$-algebras be "noncanonically" isomorphic?
Is there a locally compact group $G$ such that the canonical map from $C^{*}(G)$ to $C^{*}_{red} G$ is not isomorphism, hence $G$ is not amenable but these two $C^{*}$ algebras are isomorphic ...
- 356
5
votes
1
answer
250
views
Function algebra of Furstenberg boundary $\partial_F \Gamma$: when is it a $W^*$-algebra?
Let $\Gamma$ be a non-amenable discrete group and consider its Furstenberg boundary $\partial_F \Gamma$. It is known that this is a compact topological space which is stonean (equivalently: extremely ...
- 525
5
votes
0
answers
265
views
Failure of Tomiyama's property ($F$) for reduced group $C^*$-algebras
Are there known examples of discrete groups such that the minimal tensor product of their reduced group $C^\ast$-algebras does not have Tomiyama's property ($F$)?
Such groups must necessarily be non-...
- 51
5
votes
0
answers
270
views
A generalization of real characters on a group
Yesterday I understood that I can't live without this construction:
Let $G$ be a group, $A$ an associative algebra over $\mathbb R$ and $n\in{\mathbb N}$. We consider a sequence of maps $\varphi_k:...
- 7,426
5
votes
0
answers
341
views
Is translation by the free group (in two generators) on a certain completion of the group an amenable action?
Let $\mathbb{F}_2 = \langle a,b\rangle$ be the free group in two generators $a,b$ and let $\alpha \in \text{End}(\mathbb{F}_2)$ be given by $\alpha(a) = a^2, \alpha(b)= b^2$. Note that the index $[\...
- 188
4
votes
1
answer
261
views
Uniform Roe algebra of virtually abelian group is type I C*-algebra?
Let $G$ be an arbitrary (discrete) group. It acts by left translation on $\ell^\infty(G)$. The uniform Roe algebra of $G$ is defined as the crossed product $\ell^\infty (G) \rtimes_{\mathrm{red}}G$.
...
- 741
4
votes
1
answer
199
views
Groups for which all projections of $C^*_{\text{red}}G$ belong to $\mathbb{C}G$
Revision: According to comment of Wojowu we give a complete revise for this post.
A group $G$ is a pr-group if all projections of $C^*_{\text{red}} G$ are contained in its dense subalgebra $\mathbb{...
- 356
4
votes
1
answer
1k
views
Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra
Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$.
Let $p,q \in M_{\infty}(A)$ be (...
4
votes
0
answers
220
views
Bochner theorem for (non-abelian) discrete groups
I am interested in Pontryagin duality-like theories for discrete groups, more particularly, whether an analogue to Bochner's theorem for abelian groups exists in the discrete non-finite and non-...
- 261
3
votes
1
answer
223
views
representations with centralizer stable under conjugate transpose
Let $\rho:G\to GL_n(\mathbb{C})$ be a finite-dimensional representation of a finite group $G$ over $\mathbb{C}$, and $C_\rho\subset M_n(\mathbb{C})$ its centralizer, i.e. $m\in C$ iff $m$ commutes ...
- 14k
3
votes
1
answer
285
views
Extreme points of the set of all traces
Let $G$ be a finitely generated group with a bound on its complex unitary irreducible representations: That is assume all complex unitary irreducibles of $G$ have degrees at most $k$ for some integer $...
- 193
2
votes
1
answer
307
views
When the reduced $C^*$-algebra of $\Gamma$ admits character then $\Gamma$ is amenable [closed]
Suppose that $C^*_r(\Gamma)$ admits some character (homomorphism into $\mathbb{C}$)-here $\Gamma$ is discrete group and $C^*_r(\Gamma)$ is the closure of the image of the group ring $\mathbb{C}\Gamma$ ...
- 9,330
2
votes
1
answer
99
views
Definite negative functions and length functions
$\DeclareMathOperator\ND{ND}$I am reading E. Bedos paper on heat properties for groups.
Let's denote, for a group G, $$\ND^+_0(G) := \{d : G \to [0,+\infty[\; : \;d \text{ is negative definite and }d(...
- 21
1
vote
1
answer
89
views
Continuous functions on HLS groupoids
I am reading a paper about property (T) for groupoids: Topological property (T) for groupoids. In section 4.4 they discuss the HLS groupoids which I describe define here.
Let $\Gamma$ be a discrete ...
- 261
1
vote
0
answers
86
views
A cross product on $C^*_{red} G$
For every group $G$, the reduced group $C^*$-algebra $C^*_{red}G$ is equipped with the inner product $\langle a,b\rangle=tr(ab^*)$ where "$tr$" is the standard trace on group $C^*$-algebras.
For ...
- 356
0
votes
1
answer
204
views
A certain class of representations
Let $g$ be a non-identity element in a torsion-free amenable group, does there exist a finite-dimensional unitary representation $\pi$ with $\pi(g)\neq 1$?
(The word "finite-dimensional" was ...
- 2,118
0
votes
0
answers
373
views
Amenability of an "almost Hamiltonian" group
Here is another interesting question that I can't answer on my own.
Let $G$ be a countable, discrete group such that for any subgroup $H$ of $G$ and any element $s$ of $G$ we have $[H : sHt]$ is ...
